Calviño Louzao, EstebanGarcía Río, EduardoVázquez Lorenzo, RamónGutiérrez Rodríguez, Ixchel Dzohara2019-08-012019-08-012019http://hdl.handle.net/10347/19468Einstein manifolds, being critical for the Hilbert-Einstein functional, are central in Riemannian Geometry and Mathematical Physics. A strategy to construct Einstein metrics consists on deforming a given metric by a conformal factor so that the resulting metric is Einstein. In the present Thesis we follow this approach with special emphasis in dimension four. This is the first non-trivial case where the conformally Einstein condition is not tensorial and there are topological obstructions to the existence of Einstein metrics. The conformally Einstein condition is given by a overdetermined PDE-system. Hence the consideration of necessary conditions to be conformally Einstein are of special relevance: the Bach-flat condition is central. In this Thesis we classify four-dimensional homogeneous conformally Einstein manifolds and provide a large family of strictly Bach-flat gradient Ricci solitons. We show the existence of Bach-flat structures given as deformations of Riemannian extensions by means of the Cauchy-Kovalevskaya theorem.engAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Bach tensorConformally Einstein manifoldsRicci solitonsMaterias::Investigación::12 Matemáticas::1204 Geometría::120404 Geometría diferencialMaterias::Investigación::12 Matemáticas::1204 Geometría::120411 Geometría de Riemanndoctoral thesisopen access